Vom Fachbereich Elektrotechnik und Informationstechnik der Technischen Universit¨at Darmstadt
zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.)
genehmigte Dissertation
von
Pouyan Parvazi, M.Sc.
Geboren am 21. M¨arz 1975 in Tehran, Iran.
Referent: Prof. Dr.-Ing. Marius Pesavento
Korreferent: Prof. Dr.-Ing. Christoph F. Mecklenbr¨auker Tag der Einreichung: 8. Dezember, 2011
Tag der m¨undlichen Pr¨ufung: 18. Januar, 2012
D17
Darmst¨adter Dissertation 2012
Minoo Mobassery
and Ebrahim Parvazi,
for their unbounded and unconditional love.
I would like to express my deepest gratitude to: Alex Gershman
who made all this possible. May he rest in peace. Marius Pesavento
who is my supervisor, my colleague, and my friend; all in one. For his patience and for his invaluable advices.
Christoph Mecklenbr¨auker for his time and support. Marlis Gorecki
for all her kindness and help.
Adrian, Ahmed, Christian, Imran, Michael, Nils, Nima, and Philipp
and all my colleagues and friends in NTS group for all their support and help, and for all the discussions and laughs.
and ... Shadi
my love and my joy.
Die vielf¨altigen Anwendungen der Sensorgruppensignalverarbeitung und ihre spezifischen Herausforderungen in der Praxis bilden die Motivation f¨ur die vorliegende Arbeit. Schwierig-keiten, die unter nicht-idealen Bedingungen auftreten und in dieser Arbeit betrachtet werden sind: eine begrenzte Anzahl von verf¨ugbaren Schnappsch¨ussen oder eine niedrige Signal-leistung, eine ungenaue Kenntnis der Sensorgruppenanordnung und das Fehlen bestimmter zeitlicher oder raumlicher Abtastwerte. Diese Bedingungen werden bei der Sch¨atzung der folgenden Parameter ber¨ucksichtigt: die Einfallsrichtung (DOA) der Signale auf die Sen-sorgruppe, die Array-Mannigfaltigkeit, und die Frequenz und die D¨ampfungsfaktoren der Harmonischen des Signals.
Um praktische Gegebenheiten wie die begrenzte Anzahl von Schnappsch¨ussen oder eine geringen Signalleistung zu bew¨altigen wird zun¨achst eine Methode eingef¨uhrt, die auf der Idee einer Sch¨atzer-Bank und der Detektion und Korrektur fehlerhafter Sch¨atzwerte basiert. Das vorgeschlagene Verfahren verbessert die G¨ute der DOA-Sch¨atzung signifikant in Szenar-ien, in denen einige der Quellen eng beieinander liegen.
Dann werden Unbestimmtheiten bez¨uglich der relativen Lage der Sensorgruppen be-trachtet. Der Fokus dieses Teils der Untersuchung liegt auf der Richtungssch¨atzung sowie auf der Sensorgruppenkalibrierung bei teilkalibrierten Sensorgruppen. Drei verschiedene Typen teilkalibrierter Sensorgruppen werden untersucht: die allgemeinste Form einer teil-kalibrierten Sensorgruppe, die Anordnung eines teilweise teil-kalibrierten Arrays bestehend aus mehreren identischen Untergruppen und Anordnung einer Sensorgruppe bestehend aus paarweise kalibrierten Sensoren. Die hochaufl¨osenden Verfahren zur blinden Sensor-gruppenkalibrierung und simultanen Richtungssch¨atzung, die f¨ur jede dieser Sensorgrup-penanordnung vorgeschlagen werden, weisen eine erheblich verbesserte Sch¨atzgenauigkeit
der Frequenz- und D¨ampfungsparameter einer Superposition von mehreren Harmonischen betrachtet. Dieses Problem kann als Verallgemeinerung des zuvor betrachteten Richtung-sch¨atzproblems mittels Sensorgruppen betrachtet werden. Es werden suchfreie und doch hochaufl¨osende Verfahren zur eindeutigen Sch¨atzung der Signalparameter vorgestellt, die sich durch exzellente Sch¨atzgenauigkeit auszeichnen.
The variety of sensor array processing applications and their practical concerns are the motivations behind the present thesis. The uncertainties in the non-ideal conditions con-sidered in this thesis are: the limited number of available snapshots or low signal power, uncertainties in the sensor array geometry, and the nonavailability of specific temporal or spatial samples. These conditions are taken into account in the estimation process of the following parameters: the direction-of-arrival (DOA) of the signals impinging on the array, the array manifold, and the frequencies and the damping factors of the signal harmonics.
First, to deal with the practical situations of a limited number of snapshots or a low signal power, a method is introduced based on the estimator bank approach combined with detection and cure of erroneous estimates. The proposed technique significantly improves the DOA estimation performance in the scenarios where some sources are closely-spaced.
Next, uncertainties in the array sensor locations are considered. The focus of this part of the study lies on the blind calibration and joint DOA estimation in partly-calibrated arrays. Three types of calibrated array geometries are investigated: the arbitrary partly-calibrated array, the partly-partly-calibrated array composed of multiple identical subarrays, and the pairwise-calibrated array. The novel high-resolution DOA and array manifold estimation techniques proposed for each of these array types demonstrate superior DOA estimation performance in comparison with the state-of-the-art methods.
Lastly, the nonavailability of specific samples in the harmonic retrieval problem, i.e., the problem of estimating the frequencies and the damping factors of a harmonic mixture, is considered. The harmonic retrieval problem can be regarded as the generalization of the DOA estimation problem. In the case of incomplete samples, search-free yet high-resolution techniques are proposed which demonstrate excellent harmonic estimation performance.
Acknowledgments iii Zusammenfassung v Abstract vii Table of Contents ix List of Figures xi List of Tables xv Acronyms xix Notation xxi 1 Introduction 1 2 Signal Model 7
3 Previous Works and the State-of-the-Art 19
3.1 Introduction . . . 19
3.2 Multiple Signal Classification (MUSIC) and Its Variants . . . 20
3.2.1 MUSIC . . . 20
3.2.2 Weighted-MUSIC . . . 21
3.2.3 Root-MUSIC . . . 22
3.3 Rank-Reduction (RARE) Method . . . 24
3.4 Multiple Invariance MUSIC (MI-MUSIC) . . . 26
3.5 Estimation of Signal Parameters via Rotational Invariance Techniques (ES-PRIT) . . . 26
3.6 Generalized ESPRIT (GESPRIT) . . . 28
3.7 The Missing Data Iterative Adaptive Approach (MIAA) . . . 29 ix
4.3 Detecting Outlying Estimates Using Hypothesis Testing . . . 37
4.3.1 The Generalized Likelihood Ratio Test (GLRT) . . . 39
4.3.2 The Locally Most Powerful Test (LMPT) . . . 41
4.4 Outlier Identification and Cure (I&C) . . . 42
4.5 Combination of the Cured Sets of Estimates . . . 44
4.5.1 Method Based on Cluster Finding . . . 46
4.5.2 Method Based on Deadzone-Linear Function . . . 47
4.6 Simulations . . . 49
4.7 Chapter Summary . . . 55
5 Joint DOA Estimation and Array Manifold Calibration in Partly-Calibrated Arrays 63 5.1 Introduction . . . 63
5.2 Arbitrary Partly-Calibrated Array (APCA) . . . 66
5.2.1 Signal Model for APCA . . . 66
5.2.2 Methods for Estimating the Displacement-Phase Matrix . . . 70
5.2.3 DOA Estimation Using a Multivariate Function . . . 71
5.2.4 DOA Estimation Using a Univariate Function . . . 74
5.3 Partly-Calibrated Array Composed of Multiple Identical Subarrays (PCAMIS) 80 5.3.1 Signal Model for PCAMIS . . . 80
5.3.2 Indirect Estimation of the Manifold Vector . . . 83
5.3.3 Direct Estimation of the Manifold Vector . . . 85
5.3.4 Convex Optimization Approach for Estimating the Manifold Vector 89 5.4 Pairwise-Calibrated Arrays (PWCA) . . . 91
5.4.1 Signal Model for PWCA . . . 92
5.4.2 DOA Estimation Method for PWCA . . . 93
5.5 Simulations . . . 96
5.6 Chapter Summary . . . 108
6 Fast Algorithms For Harmonic Retrieval (HR) Problem 109 6.1 Introduction . . . 109
6.2 Signal Model for Harmonic Retrieval Problem . . . 112
6.3 The Weighted Multiple Invariance (WMI) Approach . . . 114
6.4 The Complete Sample Case . . . 118
6.5 The Incomplete Sample Case: Methods Based on Selection . . . 120
6.5.1 Selection Based on the Residual Polynomial . . . 121
6.5.2 Selection Based on the MUSIC Criterion . . . 122
6.6 The Incomplete Sample Case: Method Based on Polynomial Intersection . . 123 x
6.6.4 Computational Complexity . . . 128
6.7 Simulations . . . 129
6.8 Chapter Summary . . . 134
6.9 Appendices . . . 140
7 Conclusions and Future Works 147
Bibliography 151
2.1 Sensor array model . . . 8 2.2 Scenario II (APCA geometry): arbitrary known subarrays with arbitrary
unknown displacements . . . 10 2.3 Scenario III (PCAMIS geometry): identical arbitrary known/unknown
sub-arrays with arbitrary unknown/known displacements . . . 11 2.4 Scenario VI (PWCA geometry): two arbitrary unknown subarrays with
ar-bitrary known displacements between sensor pairs . . . 12 2.5 PWCA geometry and its relation to APCA geometry . . . 13 2.6 Uniform linear array (ULA) model . . . 15 3.1 ESPRIT geometry: two arbitrary subarrays with known intersubarray
dis-placement vector . . . 27 4.1 Illustration of estimator bank method combined with outlier I&C. . . 50 4.2 Histogram of γ0( ˜R) and the scatter plot of the RMSEs of the
MUSIC-generated DOA estimates versus γ0( ˜R) for SNR = 0dB. . . 55
4.3 Histogram of γ0( ˜R) and the scatter plot of the RMSEs of the
MUSIC-generated DOA estimates versus γ0( ˜R) for SNR = 3dB. . . 56
4.4 Histogram of γLM P T( ˜R) and the scatter plot of the RMSEs of the
MUSIC-generated DOA estimates versus γLM P T( ˜R) for SNR = 3dB. . . 56
4.5 RMSE versus SNR for the GLRT quality assessment method and Pκ= 0.5 . 57
4.6 RMSE versus sample number for the GLRT quality assessment method and Pκ= 0.5, SNR= 6dB . . . 57
4.7 RMSE versus SNR for the LMPT quality assessment method and Pκ = 0.5 58
4.9 Distribution of RMSEs before and after the I&C for different threshold values 59
4.10 Illustration of the cluster finding method with random data points . . . 59
4.11 Illustration of the cluster finding method with simulated DOA estimates from estimator bank at SNR= −7dB data points . . . 60
4.12 RMSE vs. SNR for Pκ = 0.5 and the GLRT method . . . 60
4.13 RMSE vs. SNR for Pκ = 0.95 and the GLRT method . . . 61
4.14 RMSE vs. SNR for Pκ = 0.95 and the LMPT method . . . 61
4.15 DOA estimation RMSEs versus number of snapshots for Pκ = 0.95 and the GLRT method . . . 62
4.16 DOA estimation RMSEs versus number of snapshots for Pκ = 0.5 and the GLRT method . . . 62
5.1 DOA estimation performance (RMSE) vs. SNR for Simulation Setup 1 . . . 101
5.2 DOA estimation performance (RMSE) vs. SNR for Simulation Setup 1 . . . 102
5.3 Resolution probability vs. SNR for Simulation Setup 1 . . . 102
5.4 Resolution probability vs. SNR for Simulation Setup 1 . . . 103
5.5 DOA estimation performance (RMSE) vs. SNR for Simulation Setup 2 . . . 103
5.6 DOA estimation performance (RMSE) vs. SNR for Simulation Setup 2 . . . 104
5.7 Resolution probability vs. SNR for Simulation Setup 2 . . . 104
5.8 Resolution probability vs. SNR for Simulation Setup 2 . . . 105
5.9 DOA estimation performance (RMSE) vs. SNR for Simulation Setup 3 . . . 105
5.10 Resolution probability vs. SNR for Simulation Setup 3 . . . 106
5.11 First subarray manifold estimation performance (RMSE) vs. SNR for Simu-lation Setup 3 . . . 106
5.12 DOA estimation performance (RMSE) vs. SNR for Simulation Setup 4 . . . 107
5.13 Resolution probability vs. SNR for Simulation Setup 4 . . . 107
6.1 Performance comparison of different weighting coefficients for the damped HR complete sample case (|µ| < 1). . . 134
6.3 Performance comparison of different weighting coefficients and MIAA for the undamped HR complete sample case (|µ| = 1). . . 135 6.4 Performance comparison of different methods for the damped HR incomplete
sample case (|µ| < 1). . . 136 6.5 Resolution probability vs. SNR for different methods in the damped HR
incomplete sample case (|µ| < 1). . . 136 6.6 The effect of iteration in the selection of the weighting coefficients on the
performance of the proposed method for the damped HR incomplete sample case . . . 137 6.7 Performance comparison of different methods for the undamped HR
incom-plete sample case (|µ| = 1). . . 137 6.8 Resolution probability vs. SNR for different methods in the undamped HR
incomplete sample case (|µ| = 1). . . 138 6.9 Performance comparison of different methods for the undamped HR
incom-plete sample case (|µ| = 1). . . 138 6.10 Resolution probability vs. SNR for different methods in the undamped HR
incomplete sample case (|µ| = 1). . . 139
3.1 Modified MIAA for multiple snapshots . . . 30
4.1 Algorithm EBQA (for the GLRT) . . . 48
4.2 Algorithm EBIC (for the GLRT) . . . 49
5.1 AP Algorithm for h(θ) . . . 72 5.2 Algorithm APCA-I . . . 74 5.3 Algorithm APCA-II . . . 77 5.4 Algorithm APCA-III . . . 79 5.5 Algorithm PCAMIS-I . . . 86 5.6 Algorithm PCAMIS-II . . . 88 5.7 Algorithm PWCA . . . 95 6.1 Algorithm HR-I . . . 120 6.2 Algorithm HR-II . . . 122 6.3 Algorithm HR-III . . . 123 6.4 Algorithm HR-IV . . . 127 xvii
APCA arbitrary partly-calibrated array AR auto-regressive
CDF cumulative distribution function CRB Cramer-Rao bound
DOA direction-of-arrival EB estimator bank
ESPRIT estimation of signal parameters via rotational invariance technique FB forward/backward averaging
FOV field-of-view
GESPRIT generalized ESPRIT
GLRT generalized likelihood ratio test HR harmonic retrieval
IAA iterative adaptive approach I&C identification and cure LMPT locally most powerful test LR likelihood ratio
LS least squares LSE least squares error LTE long-term evolution MA moving-average
MI multiple invariance
MIAA missing data iterative adaptive approach MI-MUSIC multiple invariance MUSIC
ML maximum likelihood
MLSE minimum least squares error MODE method of DOA estimation MP matrix polynomial
MUSIC multiple signal classification NRMSE normalized root-mean-square error NMR nuclear magnetic resonance
PCA partly-calibrated array
PCAMIS partly-calibrated array composed of multiple identical subarrays PDF probability density function
PWCA pairwise-calibrated array QA quality assessment RARE rank-reduction method RMSE root-mean-square error SNR signal-to-noise ratio ULA uniform linear array UCA uniform circular array UMPT uniformly most powerful test WLS weighted least squares WMI weighted multiple invariance WMUSIC weighted-MUSIC
[·](m) the m-th entry of a vector
[·](m,l) the entry in the m-th row and the l-th column of a matrix (·)∗ conjugate of a complex variable
(·)T Transpose of a matrix (·)H Hermitian of a matrix
(·)† Moore-Penrose pseudo-inverse of a matrix | · | absolute value of a complex variable k · k Frobenius norm of a matrix
](·) phase of a complex variable
D{·} operator replacing the off-diagonal entries of a square matrix by zeros and constraining the diagonal entries to unit magnitude
CN complex normal distribution CW complex Wishart distribution det{·} determinant of a matrix
diag{·} main diagonal of a matrix or a diagonal matrix with the inside vector as the main diagonal
diag+{·} a diagonal matrix constructed from the positive diagonal entries of another matrix while replacing the non-positive diagonal and all the off-diagonal entries with zeros
E{·} statistical expectation
Im{·} imaginary part of the entries of a complex matrix xxi
N {·} null-space of a matrix O order of complexity R{·} range-space of a matrix rank{·} column-rank of a matrix
Re{·} real part of the entries of a complex matrix Res(f, g) resultant of two polynomials f and g tr{·} trace of a matrix
Vmin{·} minor eigenvector of a matrix
a array manifold vector A array manifold matrix B the set of random estimators
C matrix depending on the masking indicators v and the parametric generator µ in the WMI method
em the m-th column of an identity matrix
Em,l a matrix containing a one in the (m, l) entry and zeros elsewhere
G(µ) MP containing signal subspace matrix which drops rank when its variable is equal to the true harmonics
H0 null hypothesis
H1 alternative hypothesis
h rows of the signal generator matrix in the HR method I identity matrix
Jr number of Monte-Carlo runs
KR the block diagonal matrix containing the known part of the manifold vectors in
RARE method
K number of the subarrays L number of the signals
M number of array sensors or number of samples in HR problem xxii
Mk number of sensors in the k-th subarray
M(µ) MP containing signal generator matrix which drops rank when its variable is equal to the true harmonics
n noise vector
N number of snapshots
Ng number of search grid points in a search-based method
P mixing matrix
Q estimator bank size or number of random estimators rs,l the l-th source power
ˆ
rs,l estimated l-th source power
ˆrs estimated source power vector
R received signal covariance matrix ˆ
R sample received signal covariance matrix ˜
R reconstructed received signal covariance matrix Rs signal covariance matrix
s signal vector t time index
T set of estimates from all the random estimators in an estimator bank UN noise subspace matrix
US signal subspace matrix
V diagonal matrix consisting of binary masking indicators in HR v binary masking indicator in HR
¯
v available sample positions w weight vector
W weight matrix
[xm ym]T location of the m-th sensor in the xy plane
z received signal vector
γ0 quality assessment value
δi,k Kronecker delta function (equal to one if i = k and zero otherwise)
ε step size in FOV scanning ηk the k-th displacement vector θl DOA of the l-th signal
θ DOA vector
ˆ
θ DOA estimate vector κ assigned threshold value λ signal wavelength
λm received signal covariance matrix eigenvalue
ΛN noise eigenvalue matrix
ΛS signal eigenvalue matrix
µ unknown signal-dependent parameters Π projector matrix
Π⊥ orthogonal projector matrix
ρ estimated parameter set ρ = {ˆθ, ˆrs, ˆσ2}
σ2 noise power ˆ
σ2 estimated noise power
Φ displacement-phase matrix ϕ roots in root-MUSIC method
Ψ unknown array geometry-dependent parameter matrix ω frequency harmonic in the MIAA
ω vector containing the real and the imaginary parts of the true harmonics Ω matrix containing the derivatives of signal generator vectors with respect
to each generator
Introduction
Sensor array signal processing is a well-established yet still active research area of interest in signal processing. The objective of sensor array processing is the estimation of the signal parameters generated by some emitters (sources) making use of snapshots of the received signals at the output of the sensors located at different points. These parameters consist of (but are not limited to) the number of sources, the signal powers, the signal waveforms, the signal polarizations, the source velocities, the signal frequencies, the distances of the sources from the array in both the near-field and the far-field cases, and the direction-of-arrival (DOA), i.e., the azimuth and the elevation angles of the signals. Estimating the array parameters such as the spatial signature or the array manifold through sources at known or unknown locations, i.e., array calibration, is yet another crucial problem, e.g., for DOA estimation, in the array processing.
Parameter estimation in array processing plays a significant role in many diverse appli-cations. In radar, the phased antenna arrays are used both in active (as transmitters) and in passive (as receivers) modes for estimating parameters such as velocity, range, and DOA of the target objects. In sonar, the sensor arrays are used for similar purposes as in radar with the major difference that acoustic waves are used in water instead of electromagnetic waves in the atmosphere. Hence, the type of the sensors (hydrophones in the case of sonar, antennas in the case of radar) and the array design become different. Another application for antenna arrays is radio astronomy where the passive sparse array aperture covers large areas and the array is used to record images of a certain region of the sky and to estimate the
characteristics of astronomical objects such as pulsars and galaxies. In wireless communica-tions, the smart antenna arrays, i.e., adaptive arrays, are used in multi-user communication environment for applications such as beamforming in both active and passive modes, and estimating the propagation delays in a multiuser asynchronous environment. In seismology, arrays of geophones are used, e.g., to acquire information about the earth layers for oil exploration, or to study earthquakes. Array processing has also found its way into medical applications such as brain activity localization and tomography, as well as into industrial applications such as automatic monitoring, fault detection, and localization.
The variety of the applications and their particular requirements are the motivations behind the present thesis. Our focus lies on the process of the estimation of the signal pa-rameters (signal DOAs or signal frequencies and damping factors) and of the estimation of array manifold. In this thesis, difficult and non-ideal conditions and uncertainties are con-sidered in the sensor array processing. By difficult conditions we are referring specifically to limited number of available snapshots and low power signals which make the estimation of the signal DOAs difficult. Robust estimators for the sparse sensor arrays is another subject of the present thesis. In the sparse sensor array model where the array is composed of several subarrays, uncertainties in the time synchronization between subarrays, the fading for different subarrays, and inter-subarray displacements may lead to a severe performance degradation of conventional DOA estimators. Therefore, robust estimation techniques able to properly estimate the signal parameters despite those uncertainties are of practical im-portance. Uncertainties like corrupt samples are considered for the case of signal frequency (and its damping factor) estimation in the harmonic retrieval problem.
The following ideas and techniques are presented in this thesis to overcome such difficult and non-ideal conditions:
In practical situations where the number of snapshots is limited or the SNR is low, the DOA estimation performance of subspace-based methods degrades substantially [29], [95]. To mitigate such performance degradations, several methods have been proposed in [2], [7], [17], [35], [46], [52], [73]. In Chapter 4, a method is introduced based on the concepts of “estimator bank” [17] and of the detection and cure of erroneous estimates, i.e., outliers
[1], [2]. For the purpose of outlier detection and cure, hypothesis testing is used. Two ways for computing the test statistics are presented. The proposed techniques based on the estimator bank approach, as we shall see, significantly improve the DOA estimation performance in the scenarios where the number of snapshots is small, the signal powers are low, and the sources are closely-spaced. Moreover, two methods to combine the sets of estimates obtained from the estimator bank are proposed. The following publications report the results of this chapter:
• P. Parvazi, A. B. Gershman, Y. I. Abramovich, “Detecting outliers in the estimator bank-based direction finding techniques using the likelihood ratio quality assessment,” International Conference on Acoustics, Speech, and Signal Processing (ICASSP’07), Honolulu, USA, vol. 2, pp. 1065-1068, Apr. 2007.
• P. Parvazi, A. B. Gershman, and Y. I. Abramovich, “Improving the threshold per-formance of the estimator bank direction finding techniques using outlier identifi-cation and cure”, IEEE International Symposium on Wireless Pervasive Computing (ISWPC’08), Santorini, Greece, pp. 270 - 273, May 2008.
Uncertainties in the array manifold, particularly in the sensor array geometry of large aperture arrays, are considered in Chapter 5. Due to the sensitivity of subspace-based meth-ods to such uncertainties and errors [14], [81], [90], either calibration techniques [43], [50], [51], [55], [63], [72], [79], [103] are used to obtain the complete array manifold model before estimating the DOAs, or the DOAs are estimated directly utilizing only the available infor-mation about the array model [16], [77], [80], [92]. The focus of Chapter 5 lies on joint DOA and array manifold estimation, i.e., blind calibration, in partly-calibrated arrays (PCAs). Three PCA models with proper applicability are studied: arbitrary partly-calibrated arrays (APCAs), partly-calibrated arrays composed of multiple identical subarrays (PCAMISs), and pairwise-calibrated arrays (PWCAs). The APCA and the PCAMIS models can be applied to many cases of large sparse arrays (described in more detail in Chapter 5), e.g., where the subarrays are not stationary. The PWCA model can have applications in the upcoming long-term evolution (LTE) wireless networks, where the handsets contain a pair
of antennas instead of just a single antenna. The common contribution of the different techniques introduced in this chapter is the exploitation of the estimate of the unknown or uncertain part of the array manifold and its known structure in the DOA estimation algo-rithms. The novel high-resolution DOA and array manifold estimation techniques proposed for each of these types of arrays demonstrate superior performance in comparison with the state-of-the-art methods. The results of this chapter have been presented in the following publications:
• P. Parvazi and A. B. Gershman, “Direction-of-arrival and spatial signature estimation in antenna arrays with pairwise sensor calibration,” IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’10), Dallas, TX, USA, pp. 2618-2621, Mar. 2010.
• P. Parvazi, M. Pesavento, and A. B. Gershman, “Direction-of-arrival estimation and array calibration for partly-calibrated arrays,” IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’11), Prague, Czech Republic, pp. 2552-2555, May 2011.
• P. Parvazi and M. Pesavento, “A new direction-of-arrival estimation and calibration method for arrays composed of multiple identical subarrays,” IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC’11), San Francisco, CA, USA, pp. 166-170, June 2011.
The nonavailability of some sensors in array processing applications and of some data (also referred to as samples) in the generic model of harmonic retrieval (HR) problem is considered in Chapter 6. In the harmonic retrieval model, which is the generalization of the model introduced in sensor array processing applications, the frequencies and the damping factors of a superposition of signals have to be estimated. The harmonic retrieval problem has been studied extensively for various applications [8], [12], [22], [34], [36], [37], [38], [39], [40], [42], [64], [70], [75], [89], [91], [97], [107], [110]. There are techniques for estimating the desired parameters using a search-free scheme [22], [34], [42], [64], [91], [97], [110] where
all the samples are available, and there are techniques, which consider the incomplete or missing sample case [12], [40], [89], [107]. However, the search-free methods to deal with the incomplete samples for the generic harmonic retrieval problem have not been studied. To this end, in Chapter 6, the search-free weighted multiple invariance (WMI) method [69] in the complete sample case is presented. In this case, scenario-dependent conditions are derived under which unique harmonics can be obtained. For the incomplete sample case, such conditions do not exist, hence, the proposed algorithm has to be modified and new techniques are required. One way is to select the best estimates from all the obtained estimates. Another way is to obtain the estimates directly making use of the intersection of polynomials with common factors. These fast, i.e., search-free, yet high-resolution proposed techniques, as can be seen from the simulations, can overcome the non-uniqueness issue and display efficient performance. The following publications report the results of this chapter: • P. Parvazi, M. Pesavento, and A. B. Gershman, “Exploiting multiple shift-invariances in harmonic retrieval: the incomplete data case,” IEEE Workshop on Statistical Signal Processing (SSP’11), Nice, France, pp. 729-732, June 2011.
• P. Parvazi, M. Pesavento, and A. B. Gershman, “Rooting-based harmonic retrieval using multiple shift-invariances: the complete and the incomplete sample cases,” IEEE Trans. Signal Processing, 2011, accepted.
The present thesis is organized as following:
In Chapter 2, the signal model and the subspace separation technique which is common to all the proposed methods in this thesis are introduced. In Chapter 3, the state-of-the-art and existing estimation techniques which are going to be compared with the proposed meth-ods are described. In Chapter 4, a method to improve the DOA estimation performance for fully-calibrated arrays in the cases of low SNR and small number of snapshots is proposed. In this chapter, after describing the “estimator bank” idea, the hypothesis testing concept to detect and also to cure the erroneous estimates is introduced. Then, methods to combine these two concepts to obtain an improved performance, as shown by the simulations at the end of this chapter, are proposed. In Chapter 5, novel methods for high-resolution DOA
estimation and array calibration for three types of PCAs are presented. These methods display superior performance compared to the existing methods as demonstrated by the simulations at the end of this chapter. In Chapter 6, new search-free algorithms for the complete and the incomplete sample cases of the harmonic retrieval problem are proposed. Chapter 7 concludes the thesis, and some future works and extensions of the problems studied in this thesis are also suggested for further research.
Signal Model
Consider an array with M identical omni-directional sensors as displayed in Fig. 2.1. The location of each sensor m (for m = 1, . . . , M ) is denoted by [xm, ym]T in the Cartesian
coordinate system. Depending on the scenario, the array geometry can be partitioned into smaller subarrays with completely known or partly known geometry. Assume that there are L (< M ) far-field point sources emitting narrow-band signals whose baseband model at time index t is denoted by s(t). We also assume that N observations or snapshots are available, i.e., t = 1, 2, . . . , N . Throughout the text, we have the following assumptions for the sources:
Assumption 1: The number of sources L is known or can be estimated using the well-known methods presented in [26], [101], [102].
Assumption 2: The sources are uncorrelated.
The noise in the m-th sensor for m = 1, 2, . . . , M , is modeled as independently identically distributed (i.i.d.) zero-mean complex white Gaussian additive noise, i.e.,
nm(t) ∼ CN (0, σ2). (2.1)
y x sensor 2 sensor 3 sensor 4 sensor M far-field source 1 far-field source L θ1 reference sensor
Figure 2.1: Sensor array model The noise vector n(t) is defined as
n(t) , [n1(t), n2(t), . . . , nM(t)]T (2.2)
and it is assumed to have the following statistical properties: Assumption 3: The noise is both spatially and temporally white
E{n(t1)nH(t2)} = σ2I, t1 = t2 0, t1 6= t2 (2.3) and E{n(t1)nT(t2)} = 0. (2.4)
The t-th snapshot of the array observation vector (also referred to as array output signal) in the presence of the sensor noise n(t) is given by
z(t) = L X l=1 a(µl, Ψ)sl(t) + n(t) (2.5) = A(µ, Ψ)s(t) + n(t) (2.6) where s(t) = [s1(t), s2(t), . . . , sL(t)]T (2.7)
is the signal waveform vector which is assumed to be stochastic. The M ×L matrix A(µ, Ψ) can be partitioned into L vectors, each corresponding to a particular source such that
A(µ, Ψ) , [a(µ1, Ψ), a(µ2, Ψ), · · · , a(µL, Ψ)] . (2.8)
In the sensor array processing application, the matrix A(µ, Ψ) denotes the array manifold matrix representing the directional characteristics of the array output. In a more general model which we encounter in harmonic retrieval applications, the matrix A(µ,Ψ) denotes the signal generator matrix. This case will be discussed briefly later in this section and in more details in Chapter 6. To achieve the minimum requirement for the uniqueness of the DOA estimates, we take into account the following important assumption:
Assumption 4: The matrix A(µ, Ψ) is of full-rank, i.e., rank{A(µ, Ψ)} = L.
As it can be seen, the matrix A(µ, Ψ) depends on two sets of unknown parameters. The signal-dependent parameters contained in vector µ express the received signal parameters which are the subjects of the estimation process such as the signal DOAs, frequencies or damped factors depending on the application. The second set of parameters in A(µ, Ψ), namely Ψ, denotes the source signal-independent unknowns. In the array processing appli-cations the matrix Ψ represents, for instance, the array geometry-dependent matrix. This matrix, depending on the scenario, can contain the unknown displacement vectors between the subarrays in the array or the unknown geometry of subarrays in the array. The matrix Ψ can also represent other model errors such as channel mismatches or synchronization errors. In the harmonic retrieval model, this matrix represents the unusable samples or sensors.
The scenarios that we are going to discuss can be summarized as: Scenario I: DOA estimation in fully-calibrated arrays
In this scenario the information about the arbitrary array geometry is completely known, therefore Ψ is an empty vector, and µ = θ where
θ ,[θ1, θ2, . . . , θL]T (2.9)
(reference subarray)subarray 1 subarray 2 subarray 3 subarray K
η
2η
3η
KFigure 2.2: Scenario II (APCA geometry): arbitrary known subarrays with arbitrary un-known displacements
Scenario II: DOA estimation/calibration in arbitrary partly-calibrated arrays An arbitrary partly-calibrated array (APCA) which consists of K arbitrary known subarrays is considered in this scenario as shown in Fig. 2.2. The displacement vectors between these subarrays are assumed to be unknown. Therefore µ = θ defined in (2.9) and Ψ = [η2, η3, . . . , ηK] where ηk for k = 2, . . . , K is the k-th displacement vector between the first subarray (also called the reference subarray) and the k-th subarray. By this definition the first displacement vector is η1 = 0 and, therefore, known and is not considered.
Scenario III: DOA estimation/calibration in partly-calibrated arrays composed of multiple identical subarrays
Partly-calibrated arrays with multiple identical subarrays (PCAMIS) marks a spe-cial case of APCA of Scenario II where all subarrays have identical geometry and
η2 η3 ηK subarray 1 (reference) subarray 2 subarray 3 subarray K
Figure 2.3: Scenario III (PCAMIS geometry): identical arbitrary known/unknown subar-rays with arbitrary unknown/known displacements
identically oriented as shown in Fig. 2.3. Here, without loss of generality, we assume that the displacement vectors are known. Therefore µ = θ defined in (2.9) and Ψ corresponds to the subarray geometry.
Scenario IV: DOA estimation/calibration in pairwise-calibrated array
A pairwise-calibrated array (PWCA) is defined as an array consisting of two differ-ent M1-sensor subarrays with unknown geometry and known displacement vectors
between the pairs of sensors each in one of the subarrays as shown in Fig. 2.4. This scenario can also be regarded as a special case of APCA of Scenario II where each subarray is composed of only two sensors as shown in Fig. 2.5. Therefore, µ = θ defined in (2.9) and Ψ is corresponding to the first subarray geometry.
Scenario V: Harmonic retrieval problem in the incomplete sample case
The harmonic retrieval problem is the generalization of the DOA estimation problem and the array output model for fully-calibrated uniform linear arrays (see Fig. 2.6) boils down to the model of the present scenario. In harmonic retrieval problem the
. . . subarray 1 subarray 2 η1 η2 ηM1
Figure 2.4: Scenario VI (PWCA geometry): two arbitrary unknown subarrays with arbi-trary known displacements between sensor pairs
frequencies and damping factors of a discrete harmonic mixture from one or multi-ple observations (or sammulti-ples) taken, e.g., along time, frequency or space have to be estimated. In the model of (2.6), in this case, the vector µ expresses the harmon-ics together with their possible damping factor µ = [µ1, µ2, . . . , µL]T where µl ∈ C,
|µl| ≤ 1 for l = 1, . . . , L, then A(µ) = 1 1 · · · 1 µ1 µ2 · · · µL µ21 µ22 · · · µ2L .. . ... · · · ... µM −11 µM −12 · · · µM −1L . (2.10)
In Section 6.2, we consider the case where some samples in this scenario may be un-available, due to their uncertainty or their being affected by too much noise. Therefore, the matrix Ψ demonstrates these unavailable or missing samples whose locations are assumed to be known. The location of these missing samples (or sensors in the array processing context) is defined by an M × M diagonal matrix V. Then, the matrix A in this case (see Section 6.2 for more details) can be shown to be A(µ, Ψ) = VA(µ). In the case where all the samples are available we have V = I.
. .. . .. η1 η2 ηM1 subarray 1 subarray 1 subarray 2 subarray 2 subarray M1 interchangeable geometry
Figure 2.5: PWCA geometry and its relation to APCA geometry
In DOA estimation under Scenarios I-IV, we assume that sensors are located in the xy plane with the position of the m-th sensor denoted by [xm, ym]T. Without loss of generality,
we assume that the first sensor is located in the origin (hence x1 = y1 = 0). The array
manifold matrix can be described as [33], [96], [98] A(θ) = [a(θ1), a(θ2), · · · , a(θL)]
= 1 1 · · · 1
e−j(2π/λ)(x2sin θ1+y2cos θ1) e−j(2π/λ)(x2sin θ2+y2cos θ2) · · · e−j(2π/λ)(x2sin θL+y2cos θL)
e−j(2π/λ)(x3sin θ1+y3cos θ1) e−j(2π/λ)(x3sin θ2+y3cos θ2) · · · e−j(2π/λ)(x3sin θL+y3cos θL)
..
. ... · · · ...
e−j(2π/λ)(xMsin θ1+yMcos θ1) e−j(2π/λ)(xMsin θ2+yMcos θ2) · · · e−j(2π/λ)(xMsin θL+yMcos θL)
(2.11) where the array manifold vector a(θl) for the l-th source is given by
a(θl) = 1 e−j(2π/λ)(x2sin θl+y2cos θl) e−j(2π/λ)(x3sin θl+y3cos θl) · · · e−j(2π/λ)(xMsin θl+yMcos θl) (2.12)
for l = 1, . . . , L and λ is the signal wavelength assumed to be equal for all the signals. A particular and popular array geometry is the uniform linear array (ULA) where all the array sensors lie on one axis, say x-axis, and the distance between the adjacent sensors d is identical for any two adjacent sensor (see Fig. 2.6). In general, the distance between the adjacent sensors is chosen as within half the signal wavelength d = λ/2 to avoid ambiguity in the estimates [33], [71]. The array manifold matrix for the ULAs can be written as
A(θ) = [a(θ1), a(θ2), · · · , a(θL)]
= 1 1 · · · 1
e−j(2πd/λ) sin θ1 e−j(2πd/λ) sin θ2 · · · e−j(2πd/λ) sin θL
e−j2(2πd/λ) sin θ1 e−j2(2πd/λ) sin θ2 · · · e−j2(2πd/λ) sin θL
..
. ... · · · ...
e−j(M −1)(2πd/λ) sin θ1 e−j(M −1)(2πd/λ) sin θ2 · · · e−j(M −1)(2πd/λ) sin θL
(2.13) where a(θl) = 1 e−j(2πd/λ) sin θl e−j2(2πd/λ) sin θl · · · e−j(M −1)(2πd/λ) sin θl (2.14)
for l = 1, . . . , L. As it can be observed the obtained array manifold matrix for the ULAs exhibits the Vandermonde structure [21]. The popularity of the ULAs is due this structural feature which has been exploited to develop search-free, low computational complexity DOA estimation algorithms [7], [11], [19], [77], [88].
d d
d
x
Figure 2.6: Uniform linear array (ULA) model
The covariance matrix of the array output signal for all the five scenarios is defined as R , E{z(t)zH(t)}
= A(µ, Ψ)E{s(t)sH(t)}AH(µ, Ψ) + E{n(t)nH(t)}
= A(µ, Ψ)RsAH(µ, Ψ) + σ2I (2.15)
in which it is assumed that the noise and the signals are independent and have zero-mean. Moreover, in (2.15), we define the L × L signal covariance matrix as
Rs , E{s(t)sH(t)} (2.16)
where besides Assumption 2, the following assumption is also taken into account: Assumption 5: The matrix Rs is of full-rank, i.e., rank{RS} = L.
The diagonal entries of Rs indicate the power of the signals and are denoted by positive
parameters rs,1, rs,2, . . . , rs,L. From Assumptions 4 and 5, we can write
Rs= diag{rs,1, rs,2, . . . , rs,L}. (2.17)
Therefore, since from Assumption 4 the M × L array manifold matrix A(µ, Ψ) is of full-column rank, the M × M matrix A(µ, Ψ)RsAH(µ, Ψ) is of rank L (< M ), and, therefore,
rank-deficient. This low-rank property can be exploited in the presence of the sensor noise to identify two complementary subspaces, i.e., the signal- and the noise subspace, respectively. These two subspaces are at the foundation of subspace-based estimation methods. From (2.15) it can be observed that R has M − L eigenvalues equal to the noise power σ2 and L eigenvalues greater than σ2. In other words, if we define λm as the m-th largest eigenvalue
of R then
λ1≥ λ2 ≥ · · · ≥ λL> λL+1 = · · · = λM = σ2. (2.18)
Hence the L largest eigenvalues of R are called signal eigenvalues and the M − L smallest eigenvalues are called noise eigenvalues. After performing the eigen-decomposition on the array output covariance matrix R we obtain
R = M X m=1 λmumuHm = USΛSUHS + UNΛNUHN = USΛSUHS + σ2UNUHN (2.19)
where ΛS and ΛN denote, respectively, the L × L and the (M − L) × (M − L) diagonal
matrices containing the signal and the noise eigenvalues, i.e.,
ΛS = diag{λ1, λ2, . . . , λL} (2.20)
ΛN = σ2I. (2.21)
Moreover, the M × L signal- and the M × (M − L) noise eigenvector matrices US and
UN, respectively, contain the eigenvectors corresponding to the signal and to the noise
eigenvalues. We will refer to the matrices US and UN as signal subspace matrix and noise
subspace matrix, respectively, for the reasons that become apparent later on.
It is well-known [77] that considering Assumptions 4 and 5, both the array manifold matrix and the signal-eigenvector matrix span the same subspace, i.e.,
R{US} = R{A} (2.22)
where R{·} denotes the range-space of a matrix and the dependency of A(µ, Ψ) on µ and Ψ is dropped for the sake of notational brevity. In other words, there exists an L × L full-rank matrix P, the so-called mixing matrix, such that
Similarly, since P is nonsingular, hence invertible, we can also write US = AP−1
= AP0 (2.24)
where for the simplicity in the notations in the later sections we define
P0 , P−1. (2.25)
The true array covariance matrix is generally unknown in practice, therefore, its finite sample estimate ˆ R = 1 N N X t=1 z(t)zH(t), (2.26) which is the ML estimate of the R in (2.15) in the case of Gaussian noise, is used. The following assumption is required so that the rank of the obtained sample covariance matrix
ˆ
R (in the presence of the noise) becomes equal to M .
Assumption 6: The number of snapshots is larger than the number of sensors, i.e., N ≥ M .
This assumption is a necessary condition for the subsequent construction of the signal and noise subspaces for the subspace-based methods which are discussed in following chap-ters. It is worth mentioning that for the ULAs, in the case of low-rank sample covariance matrix, e.g., when the sources are fully-correlated, there are forward-backward averaging (FB) and spatial smoothing techniques [41], [82] to artificially increase the number of snap-shots to obtain sample covariance matrix of higher rank.
Let ˆλm, for m = 1, . . . , M , denotes the m-th largest eigenvalue of the sample covariance
matrix ˆR in (2.26) such that ˆ
λ1 ≥ ˆλ2≥ · · · ≥ ˆλL≥ ˆλL+1≥ · · · ≥ ˆλM. (2.27)
Similar to the true covariance matrix R, the eigenvalues can be partitioned into the signal eigenvalues containing the L largest eigenvalues, i.e., ˆλ1, . . . , ˆλL, and the noise eigenvalues
consisting of the M − L smallest eigenvalues, i.e., ˆλL+1, . . . , ˆλM. Similarly, the
eigen-decomposition of the sample covariance matrix ˆR can be written as ˆ R = M X m=1 ˆ λmuˆmuˆHm = UˆSΛˆSUˆHS + ˆUNΛˆNUˆHN (2.28)
where ˆΛS and ˆΛN denote, respectively, the L × L and the (M − L) × (M − L) diagonal
matrices containing the signal and the noise eigenvalues, i.e., ˆ
ΛS = diag{ˆλ1, ˆλ2, . . . , ˆλL} (2.29)
ˆ
ΛN = diag{ˆλL+1, ˆλL+2, . . . , ˆλM}. (2.30)
The matrices ˆUS and ˆUN are, respectively, the estimates of the M × L signal- and the
M × (M − L) noise eigenvector matrices containing of the eigenvectors corresponding to the signal and to the noise eigenvalues.
Previous Works and the
State-of-the-Art
3.1
Introduction
In this chapter, we introduce some existing state-of-the-art DOA estimation and HR meth-ods. In the following chapters we compare the estimation performance of our proposed methods to the performance of the methods described in the present chapter. First, we present the well-known multiple signal classification (MUSIC) method [78] and we discuss its drawbacks. Then, the weighted-MUSIC method, which is the generalization of the MU-SIC method, and the root-MUMU-SIC method [7] are introduced. The MUMU-SIC method and its variants are only applicable to fully-calibrated arrays. For partly-calibrated arrays (PCAs), we present other DOA estimation techniques such as the rank-reduction (RARE) method [66], [68], [80], the multiple invariance MUSIC (MI-MUSIC) method [92], the estimation of signal parameters via rotational invariance techniques (ESPRIT) [77], and the generalized ESPRIT (GESPRIT) method [16], [96]. At the end, for the harmonic retrieval problem in the incomplete sample case (discussed in Chapter 6), we introduce the latest method of the missing data iterative adaptive approach (MIAA) [89]. We remark that the DOA
estimation methods can also be applied to estimate the signal harmonics.
3.2
Multiple Signal Classification (MUSIC) and Its Variants
3.2.1 MUSIC
It can be shown [78] that each column of the manifold matrix in (2.11) must be orthogonal to the noise subspace matrix UN obtained from (2.19), hence
UHNa(θl) = 0 (3.1)
for θl= θ1, . . . , θL or equivalently
aH(θl)UNUHNa(θl) = 0 (3.2)
where a(θl) is defined in (2.12). This is the core idea of the MUSIC estimation method.
In practice, in order to estimate the DOAs, the estimate of the noise subspace matrix ˆUN
obtained from the sample covariance matrix ˆR in (2.28) must be used. Therefore, the following “spectral” function is proposed in [78]
fMUSIC(θ) = 1 k ˆUH Na(θ)k2 = 1 aH(θ) ˆU NUˆHNa(θ) . (3.3)
The estimated DOAs ˆθ1, . . . , ˆθL are then obtained as the angles θ corresponding to the L
largest maxima of fMUSIC(θ) in (3.3). The denominator of the MUSIC function in (3.3) can
be interpreted as the measure of the projection of the array manifold vector onto the noise subspace ˆUN which ideally for the true DOAs is zero. Then, the estimated DOAs are the
ones that minimize this projection. To find the DOAs, a scan over the entire field-of-view (FOV) is required and the function fMUSIC(θ) in (3.3) needs to be evaluated for each θ.
The accuracy of the estimates obtained from the MUSIC spectral function in (3.3) depends on many factors such as:
• SNR
• the number of snapshots N
• the accuracy of the available array manifold vectors a(θ), itself dependent on factors such as the precision of the array and sensor calibration or the exact sensor locations Some explanation regarding these factors are in order. In order to evaluate the MUSIC function in (3.3), a limited number of scanning points should be selected, hence, a step size should be defined. If the step size is chosen small, then not only the estimation accuracy will increase but also the computational cost of the method. Therefore, some compromise must be made to have a reasonable step size (not too small) and at the same time a reasonable estimation accuracy. The effects of the value of SNR and the number of snapshots on DOA estimation of the MUSIC method will be examined in more detail later in Chapter 4. Moreover, to reduce the negative effects of these factors on the DOA estimation performance, some methods which are capable of identifying the erroneous estimates will be presented in Chapter 4. The MUSIC algorithm is known to be very sensitive to uncertainties and errors in the array manifold vectors [14], [81], [90]. This makes the exact calibration of the sensor array crucial for DOA estimation. The calibration issue in the arrays, especially large sparse arrays will be addressed in Chapter 5 and some novel techniques to simultaneously estimate the DOAs and to calibrate the sensor array will also be proposed.
3.2.2 Weighted-MUSIC
As it can be observed, in the MUSIC spectral function of (3.3), all the noise eigenvectors are treated equally. The MUSIC method can be extended to include a specific weighting matrix for controlling the effect of each noise eigenvector on the estimates. A proper choice of the weighting matrix will be particularly useful to improve the performance of the estimators in difficult situations such as low number of snapshots and low SNR to overcome some of the shortcomings of the MUSIC method [74], [98]. Toward this end, the following spectrum
function is defined to take into account the different effects of the noise eigenvectors fWMUSIC(θ) = 1 aH(θ) ˆU NW ˆUHNa(θ) . (3.4)
It is clear that the conventional MUSIC function in (3.3) is a special case of the weighted-MUSIC function in (3.4) with W = I. It should be remarked that the weighted-weighted-MUSIC function in (3.4) plays an important role in constructing the “estimator bank” in Section 4.2.
A useful choice of the weighting matrix is
W = ˆUHNe1eT1UˆN (3.5)
where e1 is the first column of the M × M identity matrix. The choice of W in (3.5)
coincides with the well-known Min-Norm method [31], [32], [74]. In the Min-Norm method, a non-zero vector with minimum norm in the noise subspace, i.e., a linear combination of the noise eigenvectors, is obtained. Then, the orthogonality of this minimum length vector and the array manifold vector is measured similar to the one used for the MUSIC method in (3.3) for the angles in the FOV. The Min-Norm method is known to yield an improved resolution capability of distinguishing two close sources, as compared to the MUSIC method in the ULAs [98].
3.2.3 Root-MUSIC
The root-MUSIC DOA estimation method [7] exploits the Vandermonde structure of the array manifold vector in the ULAs in (2.14) to estimate the DOAs through a search-free algorithm based on polynomial rooting. Defining
ϕ , e−j(2πd/λ) sin θ, (3.6) the parametric array manifold vector a(θ) becomes
a(ϕ) =1, ϕ, ϕ2, · · · , ϕM −1T
Furthermore, it is simple to show that
aH(ϕ) = aT(1/ϕ). (3.8) Then, the MUSIC criterion in (3.2) transforms into
aT(1/ϕl)UNUHNa(ϕl) = 0 (3.9)
where
ϕl= e−j(2πd/λ) sin θl (3.10)
for l = 1, . . . , L. Let us define
fr−ideal(ϕ) , aT(1/ϕ)UNUHNa(ϕ). (3.11)
From (3.8), it can be seen that if ϕ is a root of the polynomial in (3.11), then its conjugate reciprocate 1/ϕ∗ is also a root. Therefore, from (3.9), the polynomial in (3.11), which is of degree 2M − 2, has 2M − 2 roots with M − 1 roots on/inside the unit-circle and their M − 1 conjugate reciprocate pairs on/outside the unit-circle. In practice, the estimate of the noise subspace matrix, i.e., ˆUN in (2.28), from the sample covariance matrix ˆR in (2.26) is used
and the following polynomial is obtained
froot−MUSIC(ϕ) = aT(1/ϕ) ˆUNUˆHNa(ϕ). (3.12)
To estimate the DOAs, the L complex roots of froot−MUSIC(ϕ), namely ˆϕ1, . . . , ˆϕL, closest
to the unit-circle and inside it should be selected and the estimated DOAs can be computed for l = 1, . . . , L from ˆ θl= sin−1 −λ 2πd]( ˆϕl) (3.13) where ](·) denotes the phase of a complex variable. It has been demonstrated [85], [86] that both MUSIC and root-MUSIC have the same asymptotic performances. From (3.13), one can observe that the estimated DOA ˆθl (for l = 1, . . . , L) depends only on the phase of the
root ˆϕl of the root-MUSIC polynomial in (3.12) and not on the magnitude of ˆϕl. Hence, any
is robust to the radial errors of the estimated roots [98]. Because of this property, the root-MUSIC method enjoys superior performance in comparison to the root-MUSIC method in low SNR and low number of snapshots, although the root-MUSIC method is only applicable to the ULAs and also to the uniform circular arrays (UCAs) [45], and not to any arbitrary array geometry (unlike the MUSIC method). However, there are methods, such as array interpolation [15] and beamspace methods [109], in which the array manifold of an arbitrary array geometry can be approximately transformed into the array manifold of a virtual ULA so that the root-MUSIC method can be implemented.
3.3
Rank-Reduction (RARE) Method
The RARE technique has been developed in [66], [68], and [80] for the case of sensor arrays consisting of K fully-calibrated subarrays (with the total number of sensors equal to M ) without any calibration information in-between subarrays (see Fig. 2.2). This case corresponds to Scenario II or APCA model of Section 5.2. For this class of arrays, the columns of the array manifold matrix can be described as
a(θl, Ψ) = KR(θl)φR(θl, Ψ) (3.14)
where the M × K matrix KR(θl) is defined as
KR(θl) , a1(θl) 0 · · · 0 0 a2(θl) · · · 0 .. . ... . .. ... 0 0 · · · aK(θl) (3.15)
for θl = θ1, . . . , θL, ak(θl) for k = 1, . . . , K is the l-th column of the manifold matrix for
the k-th subarray such that the first sensor of that subarray is considered as the reference sensor of the corresponding subarray. The K × 1 vector φR(θl, Ψ) contains the phase
infor-mation resulting from the uncalibrated or unknown part of the array such as intersubarray displacement vectors ηk= [αk, βk]T for k = 2, . . . , K such that
where
φk(θl) , e−j(2π/λ)(αksin θl+βkcos θl) (3.17)
for k = 2, . . . , K and l = 1, . . . , L.
Note that, KR(θ) depends only on the DOAs and the known or calibrated part of the
ar-ray. The MUSIC criterion in Section 3.2.1, which exploits the property of the orthogonality of the noise subspace matrix and the array manifold matrix (3.2), can then be used
aH(θl, Ψ)UNUHNa(θl, Ψ) =
φHR(θl, Ψ)KHR(θl)UNUHNKR(θl)φR(θl, Ψ) =
φHR(θl, Ψ)FRARE(θl)φR(θl, Ψ) = 0 (3.18)
where
FRARE(θl) , KHR(θl)UNUHNKR(θl). (3.19)
The idea in the RARE algorithm is based on the observation that if K ≤ M − L, then rank{UN} ≥ K. In this case, equation (3.18) holds true only when the K × K matrix
FRARE(θl) drops rank, i.e., when rank{FRARE(θl)} < K. In the finite sample case, however,
the K × K matrix ˆFRARE(θ)
ˆ
FRARE(θ) , KHR(θ) ˆUNUˆHNKR(θ) (3.20)
is used instead. Then, in order to estimate the DOAs, the L maxima of the following function in the entire FOV must be found
fRARE(θ) =
1
|det{ ˆFRARE(θ)}|
. (3.21)
It should be remarked that the spectral-RARE function can be expressed in other ways as well, e.g., by using the minimum eigenvalue of ˆFRARE(θ) in (3.21) instead of its determinant
which yields approximately the same DOA estimation performance [80]. For specific array geometries, where the PCA is composed of identically oriented uniform subarrays, a search-free RARE algorithm, known as root-RARE, can be applied [66].
3.4
Multiple Invariance MUSIC (MI-MUSIC)
In [92], several DOA estimation techniques for arrays composed of multiple identical subar-rays (possibly with overlapping sensors) have been developed which corresponds to Scenario IV, i.e., PCAMIS geometry, in Section 5.3 (see Fig. 2.3). However, some of those methods are only applicable to a specific PCAMIS structures, for which search-free implementations exist. Since those very special array structures are not considered in this thesis, we only present here the method, namely MI-MUSIC, which is a search-based method and can be applied to the considered more general subarray model. In [92], using a subspace-fitting method [100] and assuming as in our model in Section 5.3 that there are no overlapping sensors between the subarrays, the authors in [92] developed a MUSIC function (3.2) such that
fMI−MUSIC1(θ, Ψ) = ΦMI−MUSIC(θ) ⊗ a1(θ, Ψ)HΠE⊥ˆs ΦMI−MUSIC(θ) ⊗ a1(θ, Ψ) (3.22)
where ΦMI−MUSIC(θ) , diag{φ2(θ), . . . , φK(θ)}, (3.23) φk(θ) is defined in (3.17), and Π⊥ˆ Es = I − ˆEs( ˆE H s Eˆs)−1EˆHs . (3.24)
Therefore, considering the unit-norm constraint of the manifold vectors, the DOAs can be shown to be estimated from the L minima of
fMI−MUSIC2(θ) = Lmin{(ΦMI−MUSIC(θ) ⊗ I)HΠ⊥Eˆs(ΦMI−MUSIC(θ) ⊗ I)}. (3.25)
3.5
Estimation of Signal Parameters via Rotational
Invari-ance Techniques (ESPRIT)
The ESPRIT technique is a search-free DOA estimation method applicable to arrays com-posed of two identical, possibly unknown, subarrays comcom-posed of M1 sensors with known
subarray 1 subarray 2
ηE
Figure 3.1: ESPRIT geometry: two arbitrary subarrays with known intersubarray displace-ment vector
intersubarray displacement in the direction of x-axis ηE as shown in Fig. 3.1. The array
geometry in ESPRIT technique can be regarded as either a special case of Scenario III in Fig. 2.3, where all the displacements are identical, or a special case of Scenario IV in Fig. 2.4 where the number of subarrays are limited to two subarrays.
Let us partition the signal eigenvector matrix US such that
US = US,1 US,2 . (3.26)
Following the reasoning in [77], it can be said that the rank of the M1× 2L matrix
US,1,2, [US,1, US,2] (3.27)
is L and that both US,1 and US,2 span the same subspace spanned by the manifold matrix
of the first subarray. Hence, there exists a 2L × L matrix FN= FN,1 FN,2 (3.28)
which spans the null-space of US,1,2, i.e., US,1,2FN = 0. It is shown in [77] that the
eigenvalues of the matrix −FN,1F−1N,2 contain the information about the DOAs such that
ˆ θl= sin−1 n −λ 2πηE ](L(l){−F N,1F−1N,2}) o (3.29)
where L(l){·} denotes the l-th eigenvalue of a matrix for l = 1, . . . , L.
The ESPRIT algorithm can also be used as a search-free algorithm in the fully-calibrated ULAs where the distance between two adjacent sensors is d. Let us define the two matrices A and A as the array manifold matrix with the first and the last row removed, respectively. Then, by defining US and US as the signal eigenvector matrix US with the first and the
last row removed, respectively, it can be shown that the signal DOAs can be estimated from the eigenvalues of the matrix U†SUS such that
ˆ θl= sin−1 n−λ 2πd](L (l){U† SUS}) o (3.30) for l = 1, . . . , L.
3.6
Generalized ESPRIT (GESPRIT)
The generalized ESPRIT approach of [16] has been originally formulated for the array model composed of two M1-sensor subarrays with pairwise sensor calibration such that the
displacement vectors ˜ηm = [xm − xm+M1, ym− ym+M1]
T for m = 1, . . . , M
1 between the
m-th sensor in the first subarray and its corresponding sensor, i.e., the (m + M1)-th sensor
in the second subarray, is known. This corresponds to Scenario III, i.e., PWCA geometry (see Fig. 2.4). The array geometry and the signal model are discussed in detail in Section 5.4.
It is shown in [16] that if L ≤ M1, then for any M1× L full-rank matrix W, the matrix
WHUS,2− Φp(θ)US,1
drops rank where US,1 and US,2 are defined in (3.26) and the
M1× M1 diagonal matrix Φp(θ) contains the displacement-phase information between the
sensor pairs and the M1 diagonal entries are defined as
[Φp(θ)](m,m), e−j (xm−xm+M1) sin θ+(ym−ym+M1) cos θ
(3.31) for m = 1, . . . , M1. In [16], W = US,1 has been chosen. This choice leads to the following
generalized ESPRIT spectrum [16] fGES1(θ) =
1 |det{ ˆUH
S,1UˆS,2− ˆUHS,1Φ(θ) ˆUS,1}|
where the signal DOAs are estimated from the L highest peaks of (3.32).
Another meaningful choice of W is W = US,2− Φp(θ)US,1 [96]. Then, the GESPRIT
spectral function becomes fGES2(θ) =
1
|det{ US,2− Φ(θ)US,1)H(US,2− Φ(θ)US,1}|
. (3.33) The function in (3.33) is the one that we use for simulation comparisons in Section 5.5.
3.7
The Missing Data Iterative Adaptive Approach (MIAA)
In this section, we briefly introduce the latest method in HR problem for the incomplete sample case [89]. The missing data iterative adaptive approach (MIAA) can be applied to the missing samples at arbitrary positions of a uniform sampling grid. This method has been originally proposed to recover the missing samples (or data) based on the iterative adaptive approach (IAA) [107]. However, the search-based method of MIAA can also estimate the signal frequencies, therefore, it can be used in the undamped harmonic cases. However, at the cost of two-dimensional exhaustive search, it can also be applied to the damped harmonic cases. This method uses a single snapshot and as it is shown in Section 6.7, in order to resolve multiple sources with closely-separated generators, it requires large sample size. The MIAA can be modified so that it can also perform for the case where multiple snapshots are available similar to the IAA in [107]. Here, we present the modified algorithm of the MIAA for multiple snapshots without going into the details (for more details see [89] and [107]). Since it is irrelevant to the topic in this thesis, the sample recovery part of the MIAA is not mentioned.
Suppose that the number of search grid points in the frequency domain is Ng where the
frequency parameter is defined as
ωg, 2πg/Ng (3.34)
for g = 1, . . . , Ng. Let ¯M be the total number of available samples, za(n) denotes the
Modified MIAA for Multiple Snapshots Step 1: Set Ra= I.
Step 2: Compute the complex-valued amplitude for each frequency in the grid such that ˆ s(ωg, n) = hH a (ωg)R−1a za(n) hH a(ωg)R−1a ha(ωg) (3.36) for all the points in the search grid, i.e., for g = 1, . . . , Ng and for all the snapshots,
i.e., for n = 1, . . . , N .
Step 3: Calculate the average signal power over all the snapshots at each point on the grid ˆ r(ωg) = 1 N N X n=1 |ˆs(ωg, n)|2. (3.37)
Step 4: Obtain the covariance matrix for the available samples Ra= Ng X g=1 ˆ r(ωg)ha(ωg)hHa (ωg). (3.38)
Step 5: Start from Step 2 until convergence ofPNg
g=1ˆr(ωg).
Step 6: Estimate the frequencies ˆω1, . . . , ˆωL from the L maxima of ˆr(ω) for ω =
ω1, . . . , ωNg and, consequently, µl for l = 1, . . . , L from
ˆ
µl= ej ˆωl. (3.39)
Table 3.1: Modified MIAA for multiple snapshots
n = 1, . . . , N , and ¯v1, ¯v2, . . . , ¯vM¯ be the positions where the samples are available. Defining
ha(ωg) , ejωg¯v1 ejωg¯v2 .. . ejωg¯vM¯ , (3.35)
Threshold Performance
Improvement in Fully-Calibrated
Arrays
4.1
Introduction
Sensor array processing is a rather mature field of research and many techniques have been developed to fulfill the DOA estimation task. The stochastic maximum likelihood (ML) method is known to provide the best DOA estimation performance [9] in both asymptotic and non-asymptotic regions; an ideal property which its counterpart, deterministic ML, obviously lacks [86]. Nevertheless, this family of techniques are seldom applicable since the ML method contains a nonlinear optimization problem, hence a multi-dimensional search with often prohibitively high computational cost is required. The same drawback is of main concern in other similar techniques such as weighted subspace fitting method (WSF) [54]. Several papers, e.g. [11], [88] and [108], address this issue and have made attempts to lower the computational cost of stochastic ML by exploiting the array manifold structure in order to reap, although not entirely, its excellent performance property.
Another family of DOA estimation techniques is the eigen-structure-based methods also known as subspace-based methods which originated from early works as [70] and [78]. These methods are optimal for single source scenarios yet suboptimal in the case of multiple sources [108] in comparison with the stochastic ML and became widely popular mainly
because of their reduced computational complexity. MUSIC method (Subsection 3.2.1) [78] and Min-Norm method (Subsection 3.2.2) [31] utilize one dimensional search. Root-MUSIC method (Subsection 3.2.3) [7], and other subspace-based methods like ESPRIT (Section 3.5) [77] benefit from a search-free algorithm and use either rooting a polynomial or eigen-decomposition of a matrix to directly estimate the DOAs. In the case of moderate to high SNRs or adequately large number of snapshots, the performance of subspace-based techniques is close to the best achievable estimation error variance (a.k.a. Cramer-Rao lower bound or CRB for short).
However, it is a well-known fact that in the case of low SNRs or alternatively small number of snapshots, the performance becomes severely deteriorated as the SNR/number of snapshots goes below a certain level (referred to as performance threshold hereafter). This phenomena is referred to as the performance threshold effect. In other words, below a certain threshold of SNR/snapshots, the sources cannot be resolved by the estimator and severely erroneous estimates which are referred to as “outliers” are generated [3], [94]. Here, we define the resolution of an estimator as the estimator’s ability to distinguish between two closely-spaced sources. The DOA estimation performance degradation is demonstrated in the performance figures (e.g., Fig. 4.5) by the distance of the estimation performance curve from the CRB in the low SNR or small number of snapshot regions. The performance threshold is strongly dependent on the scenario, i.e., the source parameters such as location and power, and the array parameters such as sensor locations and aperture size. The threshold effect is not limited to subspace-based methods; the optimal ML method suffers from a similar performance breakdown, too. However, the ML performance breakdown occurs in much lower SNR or smaller number of snapshot regions than in the MUSIC method [2], [29], [108].
To mitigate the degrading performance threshold issue and to overcome the gap between the ML performance threshold and the MUSIC performance threshold, many researchers have analyzed the problem [24], [29], [46], [94], [95]. The breakdown in subspace-based techniques has been related to the subspace swap phenomenon. This phenomenon occurs when, due to low SNR or snapshot size effects, the estimates of the noise eigenvalues λL+1=
